Problem: Solve for $x$ : $ 4|x + 8| - 8 = -3|x + 8| + 7 $
Add $ {3|x + 8|} $ to both sides: $ \begin{eqnarray} 4|x + 8| - 8 &=& -3|x + 8| + 7 \\ \\ { + 3|x + 8|} && { + 3|x + 8|} \\ \\ 7|x + 8| - 8 &=& 7 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 7|x + 8| - 8 &=& 7 \\ \\ { + 8} &=& { + 8} \\ \\ 7|x + 8| &=& 15 \end{eqnarray} $ Divide both sides by ${7}$ $ \dfrac{7|x + 8|} {{7}} = \dfrac{15} {{7}} $ Simplify: $ |x + 8| = \dfrac{15}{7}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 8 = -\dfrac{15}{7} $ or $ x + 8 = \dfrac{15}{7} $ Solve for the solution where $x + 8$ is negative: $ x + 8 = -\dfrac{15}{7} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& -\dfrac{15}{7} \\ \\ {- 8} && {- 8} \\ \\ x &=& -\dfrac{15}{7} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $7$ $ x = - \dfrac{15}{7} {- \dfrac{56}{7}} $ $ x = -\dfrac{71}{7} $ Then calculate the solution where $x + 8$ is positive: $ x + 8 = \dfrac{15}{7} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& \dfrac{15}{7} \\ \\ {- 8} && {- 8} \\ \\ x &=& \dfrac{15}{7} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $7$ $ x = \dfrac{15}{7} {- \dfrac{56}{7}} $ $ x = -\dfrac{41}{7} $ Thus, the correct answer is $x = -\dfrac{71}{7} $ or $x = -\dfrac{41}{7} $.